We set \( \mathcal {N}_t := \lbrace (l, L)\;|\; L = (l_1,l_2,\ldots , l_r)\rbrace ,\)  where \(1\le l < l_1< l_2 < \ldots < l_r\le t\)  and \(0\le r \le t-1.\) By convention, \(L = \emptyset \)  if \(r = 0.\)  Let \(r = \ell (L)\)  be the length of \(L\) . For each the pair \((l, L)\in \mathcal {N}_t,\)  and \(1\le u < r,\)  let us denote by \(x_{(L,\,u)} := x_{l_u}^{2^{r-1} + 2^{r-2} +\, \cdots \, + 2^{r-u}}\prod _{u < d\le r}x_{l_d}^{2^{r-d}},\)  where \(x_{(\emptyset , 1)} = 1.\)  Sum [1]} defines an \(\mathbb {Z}_2\) -linear function \(\psi _{(l, L)}: P_{t-1}\longrightarrow P_t,\)  which is determined by
\( \begin{array}{ll}& \psi _{(l, L)}(\prod _{1\le j\le t-1}x_j^{a_j}) \\&= (x_l^{2^{r} - 1}\mathsf {q}_{(l,\,t)}(\prod _{1\le s\le t-1}x_s^{a_s}))/x_{(L,\,u)},\end{array}\) 
